He made his comments at the end of his 'Tonight with Vincent Browne' show on TV3 last Thursday night during a discussion about Mr Kenny's political future.
Mr Browne last night confirmed he would apologise for the comments.
"Yes it was a stupid, silly, inconsiderate remark to make and I will be apologising for it on the programme on Monday night," he said.
Mr Burke said he had got a large number of phone calls from people in Mr Kenny's hometown of Castlebar who were upset by Mr Browne's remarks. "It's unbelievable that he said that, particularly in the environment we're in and the number of suicides we have at the moment," he said.
All broadcast organisations in the state are bound by the BAI's code of conduct, which stipulates that viewers and listeners can complain about anything "which may reasonably be regarded as causing harm or offence". It can either dismiss complaints from the public or uphold them, and its findings are published on a monthly basis.
Last May, a viewer complained about comments on Mr Browne's show by journalist Nell McCafferty, who stated that she thought the Papal Cross in the Phoenix Park was a phallic symbol.
However, the BAI rejected the complaint, saying that the comment was sufficiently challenged, given the live nature of the broadcast and an immediate intervention by the presenter.